On best proximity points for set-valued contractions of Nadler type with respect to b-generalized pseudodistances in b-metric spaces

Abstract

In this paper, in b-metric space, we introduce the concept of b-generalized pseudodistance which is an extension of the b-metric. Next, inspired by the ideas of Nadler (Pac. J. Math. 30:475-488, 1969) and Abkar and Gabeleh (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107(2):319-325, 2013), we define a new set-valued non-self-mapping contraction of Nadler type with respect to this b-generalized pseudodistance, which is a generalization of Nadler’s contraction. Moreover, we provide the condition guaranteeing the existence of best proximity points for T : A → 2B. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error inf{d(x, y) : y ∈ T(x)}, and hence the existence of a consummate approximate solution to the equation T(x) = x. In other words, the best proximity points theorem achieves a global optimal minimum of the map x → inf{d(x; y) : y ∈ T(x)} by stipulating an approximate solution x of the point equation T(x) = x to satisfy the condition that inf{d(x; y) : y ∈ T(x)} = dist(A; B). The examples which illustrate the main result given. The paper includes also the comparison of our results with those existing in the literature.